Predicative Theories and Grothendieck Toposes

This project aims at applying the most recent results and techniques of topos theory to predicative Mathematics. The theoretical and applicative objectives it addresses, can be achieved by using in a substantial new way Grothendieck toposes and their theory, a possibility arisen in 2009 after the discoveries of O. Caramello. Predicative Mathematics is the branch of pure Mathematics that considers (well-founded) construction as the unique paradigm for its development, at any level of granularity: for this reason, it has a natural interpretation as the ‘Mathematics of the computable’ since, to every theorem or definition it is possible to associate a construction which can be thought of as an abstract algorithm. This project wants to characterize predicative theories in explicit topos-theoretic terms, i.e., it requires to find a formal description in the language of Grothendieck toposes to decide whether a mathematical theory is predicative, and, in the positive case, to formally associate to every theorem and definition their natural constructions. The aim is to transfer results proved in impredicative settings into the predicative world, by using the topos-theoretic description of an impredicative theory as a “bridge” toward a suitable predicative presentation. In the opposite direction, the project wants to develop a method to synthesize algorithms in impredicative frames by importing the intrinsic computational meaning of a predicative presentation. So, ultimately, this project wants to address the fundamental question whether every mathematical theory admits a predicative presentation and, if not, to what extent one can ‘approximate’ impredicative theories via predicative ones.